The pseudocircle is the finite topological space X consisting of four distinct points {a,b,c,d} with the following non-Hausdorff topology:
{ { a , b , c , d } , { a , b , c } , { a , b , d } , { a , b } , { a } , { b } , ∅ } . {\displaystyle \{\{a,b,c,d\},\{a,b,c\},\{a,b,d\},\{a,b\},\{a\},\{b\},\varnothing \}.}
This topology corresponds to the partial order a < c , b < c , a < d , b < d {\displaystyle a<c,\ b<c,\ a<d,\ b<d} where the open sets are downward-closed sets. X is highly pathological from the usual viewpoint of general topology, as it fails to satisfy any separation axiom besides T0. However, from the viewpoint of algebraic topology, X has the remarkable property that it is indistinguishable from the circle S1. More precisely, the continuous map f {\displaystyle f} from S1 to X (where we think of S1 as the unit circle in R 2 {\displaystyle \mathbb {R} ^{2}} ) given by
f ( x , y ) = { a , x < 0 b , x > 0 c , ( x , y ) = ( 0 , 1 ) d , ( x , y ) = ( 0 , − 1 ) {\displaystyle f(x,y)={\begin{cases}a,&x<0\\b,&x>0\\c,&(x,y)=(0,1)\\d,&(x,y)=(0,-1)\end{cases}}} is a weak homotopy equivalence; that is, f {\displaystyle f} induces an isomorphism on all homotopy groups. It follows that f {\displaystyle f} also induces an isomorphism on singular homology and cohomology, and more generally an isomorphism on all ordinary or extraordinary homology and cohomology theories (e.g., K-theory). This can be proven using the following observation. Like S1, X is the union of two contractible open sets {a,b,c} and {a,b,d} whose intersection {a,b} is also the union of two disjoint contractible open sets {a} and {b}. So, like S1, the result follows from the groupoid Seifert-van Kampen theorem, as in the book Topology and Groupoids. More generally, McCord has shown that, for any finite simplicial complex K, there is a finite topological space XK which has the same weak homotopy type as the geometric realization |K| of K. More precisely, there is a functor taking K to XK, from the category of finite simplicial complexes and simplicial maps and a natural weak homotopy equivalence from |K| to XK.
See also List of topologies – List of concrete topologies and topological spaces
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